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In this paper, we study exact solutions of shear flows for a viscous incompressible fluid. The proposed solutions for the velocity components are linear functions of the longitudinal coordinates. Such solutions belong to the class of Lin–Sidorov–Aristov solutions for isobaric and isothermal processes. The obtained exact solution of the Navier–Stokes equation describes a new mechanism of momentum transfer in a medium and the flow of a vertically whirling fluid. A vertical twist in a fluid arises due to the allowance for inertial forces and a nonuniform distribution of velocities at the free boundary of the fluid layer. This solution allows us to describe the counterflow of incompressible fluid in a thin layer. The condition of perfect slip on the lower solid surface of the fluid layer is considered for the obtained exact general solution. The existence of points at which the velocity field vanishes inside the fluid layer is shown. It determines the existence of stagnant points and counterflows in the fluid.

Vortex flows of a viscous incompressible fluid at constant vertical velocity under perfect slip conditions